Make the dots the color of your logo
You need to pixelate your UV coordinates which you can do as so:

x - (x modulo pixelSize) 


You can then use the same modulo nodes by normalizing their outputs to get the coordinates for your discs:

(x modulo pixelSize) / pixelSize


Create your discs the same way you have already done and use them to mix between your image texture output and a background color.



I hope this helps.